The word quadrilateral is the combination of two Latin words quadri, meaning four, and latus, meaning side. You come across quadrilaterals every day. For example, the page of a book, the top of a pencil box, the top of a dining table and so on, are all quadrilaterals (rectangular shape). 

A simple closed figure formed by joining four line segments is called a quadrilateral.  It has four sides, four angles, four vertices and two diagonals.

In this chapter, you will learn about basic properties of a quadrilateral.

In a quadrilateral ABCD :

• The four points A, B, C, D are called its vertices.
• The four line segments AB, BC, CD and DA are called its sides.
• ∠DAB, ∠ABC, ∠BCD and ∠CDA are called its angles, to be denoted by ∠A, ∠B, ∠C and ∠D respectively.
• The line segments AC and BD are called its diagonals.

Some important facts about quadrilateral:

• If each angle of a quadrilateral is less than 180°, then it is called convex quadrilateral.
• If each angle of a quadrilateral is greater than 180°, then it is called concave quadrilateral.

Angle Sum Property of quadrilateral:

• Sum of interior angles of quadrilateral is 180°

Let us consider some examples:

Example 1:

The three angles of a quadrilateral are 76°, 54° and 108°. Find the measure fourth angle.

Solution:

We know that sum of the angles of a quadrilateral is 360°.

Let the unknown angle be x

76°+  54° +  108° + x = 360°.

 x =  122°.

Example 2: The angles of a quadrilateral are in the ratio of 3 : 4 : 5 : 6.Find all its angles.

Solution:

Let the angles be 3x°, 4x°, 5x° and 6x°.

3x° + 4x° + 5x° + 6x° = 360°.

18x° = 360°.

x° = 20°.

Hence, the angles are 60°, 80°, 100° and 120°.

Example 3: The angles of a quadrilateral are in the ratio of 4 : 6 : 3. If the fourth angle is 100°, find the other angles of a quadrilateral.

Solution:

Let the angles be 4x°, 6x° and 3x°.

4x° + 6x° + 3x° + 100° = 360°.

13x° = 260°.

x° = 20°.

Hence, the angles are  80°,120° and  60°,

TYPES OF QUADRILATERAL

A closed figure with four sides is a quadrilateral. We come across many different types of quadrilaterals every day. It would be interesting to know the types of quadrilaterals, their shapes and basic properties.

Parallelogram:

A quadrilateral is called a parallelogram, if both pairs of its opposite sides are parallel. 

In the figure given below, ABCD is a quadrilateral in which:

AB ∥ DC and AD ∥ BC. 

So, ABCD is a parallelogram.

Rhombus:

A parallelogram having all sides equal is called a rhombus.

In the figure given below, ABCD is a rhombus in which:

AB ∥ DC, AD ∥ BC and AB = BC = CD = DA.

Rectangle:

A parallelogram in which each angle is a right angle is called a rectangle.

In the figure given below, ABCD is a quadrilateral in which:

AB ∥ DC, AD ∥ BC and ∠A = ∠B = ∠C = ∠D = 90°.

So, ABCD is a rectangle.

Square:

A parallelogram in which all the sides are equal and each angle measures 90° is called a square.

In the figure given below, ABCD is a quadrilateral in which:

AB ∥ DC, AD ∥ BC, AB = BC = CD = DA.

and ∠A = ∠B = ∠ C = ∠D = 90°.

So, ABCD is a square.

Trapezium:

A quadrilateral having exactly one pair of parallel sides is called a trapezium.

In the figure given below, ABCD is a quadrilateral in which AB ∥ DC. So, ABCD is a trapezium.

If non–parallel sides of a trapezium are equal, it is called an isosceles trapezium.

Kite:

A quadrilateral is called a kite if it has two pairs of equal adjacent sides but unequal opposite sides.

In the figure given below, ABCD is a quadrilateral with AB = AD, BC = DC, AD ≠ BC and AB ≠ DC.

So, ABCD is a kite.

Let us consider an example:

Example: In the square PQRS given in the figure below, PQ = 3x – 7 and  QR= x + 3 , find PS.

Solution:

As all sides are equal so, PQ = QR.

3x – 7 = x + 3.

2x =  10.

 x = 5.

PQ = 3x – 7 = 8.

QR = x + 3 = 8.

Hence PS = 8.